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We study the good shape property of boundary sections of convex solutions ofthe oblique boundary value problem for Monge-Amp\`ere equations $$\det D^2u=f(x) \text{ in } \Omega , \quad D_{\beta}u = \phi(x) \text{ on } \partial\Omega.$$ In the two-dimensional case, we prove the global $C^{2,\alpha}$estimate for the solution. When the dimension $n \geq 3$, we show that thisestimate still holds if the solution is bounded from above by a quadraticfunction in the tangent direction. We also obtain an existence result for theconvex solution of Monge-Amp\`ere equations with Robin oblique boundaryconditions.

Boundary $C^{2, \alpha}$ Regularity for the Oblique Boundary Value Problem of Monge-Amp\`ere Equations